Chemotaxis was originally described by the Keller-Segel model which was introduced in [1]}, [2]}. The model describes the characteristic movement of cells where there are two options: the cells can be attracted (move toward) by the increasing signal concentration or they can be repulsive by this signal. Afterwards, many mathematical models have been proposed to describe chemotaxis in the last few years. The simplest version, see for example [3]}, describes the competition between the diffusion, the reproduction and the nonlocal aggregation satisfying
\( \left\lbrace \begin{array}{ll}\ u_t=\Delta u-\chi \nabla \cdot (u^m \nabla c)+\lambda f(u), & x \in \Omega , t>0, \\-\Delta c+c=u^\gamma , & x \in \Omega , t>0, \\\nabla u \cdot \nu =\nabla c \cdot \nu =0, & x \in \partial \Omega , t>0, \\u(x,0)=u_0(x)\ge 0, & x \in \Omega ,\end{array}\right.\) 
